Question

How do you use L'hospital's rule to find the limit `lim_(x->oo)x^(1/x)` ?

Answer 1

The basic idea in using the rule of De l'Hospital to find indeterminate limits of powers `f(x)^(g(x))` is to rewrite it as `e^(g(x)\ln(f(x)))` and find the limit of the indeterminate product `g(x) ln(f(x))` rewriting the product as a quotient: `ln(f(x))/(1/g(x))` or `g(x)/(1/(ln(f(x)))`

If the power was indeterminate (`0^0` or `1^infty` or `infty^0`) then the obtained quotient is either indeterminate of the form `0/0` or `infty/infty`, so that the Rule of De l'Hospital applies to lift the indetermination.

In this example `x^(1/x)=e^(1/x lnx)` and `lim_{x\to infty} ln x/x=lim_{x to infty} (1/x)/1=0` by the Rule of de l'Hospital.

Thus`lim_{x \to \infty}x^(1/x)=e^0=1`

See this video on indeterminate powers for more: